An Investigatory Glance At The Transverse & Longitudinal Wave Motions

Longitudinal waves and transverse waves are all around us. The light that helps us see everything around us exhibits wave-like properties; throw a stone in a pond, and you will see waves expanding in all directions from the point of impact; radio waves, microwaves, X-rays, sound waves, tsunamis – energy from different natural and man-made phenomena emanate in the form of waves.

But what are they exactly? Are all waves the same? If not, then how do they differ? What are the different kinds of waves in existence, and what are their categories?

This write-up investigates the above questions and then focuses on analyzing the differences by comparing transverse waves vs longitudinal waves.

What Exactly Are Waves?

Well, waves are rhythmic disturbances or vibrations that propagate through a medium and transfer energy along with them without transferring matter. As we pointed out, waves are of different types, and they are all around us, from all the light and sound to the radio waves emanating from smartphones and the huge ocean breakers people surf on.

If we apply elements of deductive reasoning to wave motions evident all around us, we will note some key things à 

• Waves can transfer energy through different kinds of mediums.
• Waves can carry different kinds of energy, such as light and sound.
• Different kinds of waves travel at different speeds in different media.
• Given the wave-particle duality of light/photons, light as a wave does not need any medium to traverse and transfer energy.
• There are stark differences between a light wave and a wave in the ocean, but energy is propagating via wave motion in both cases.
• Not all energy propagates via waves, but all waves transfer the energy of one kind or another.

Waves are energy transmission processes that can be categorized as per the nature of their motion, the energy they transfer, and their need for any propagation medium.

The Different Types of Waves

A primary categorization of waves is based on their need for a medium of propagation. Matter and electromagnetic waves are the two primary wave categories via which energy is transferred in the world around us.

• Matter or mechanical waves require a specific medium to propagate and transfer energy. Waves through any medium or any matter are matter waves that propagate energy by causing some disturbance or vibration in the medium.

Vibrations cause the molecules of the medium to collide against one another, and these collisions gradually transfer energy from one point of a medium to another.

• Electromagnetic waves are starkly different from mechanical waves. Unlike matter waves, there are no oscillating particles of medium propagating energy from one point to the other; instead, we have electromagnetic fields oscillating and transferring energy.

The oscillating electromagnetic fields have different components in different directions and different phases. As you may know, variations in an electric field will induce a magnetic field in the vicinity & vice versa. Electromagnetic waves, thus, are oscillating electromagnetic fields in motion.

Where does all the energy come from? It combines all the energy stored in the mutually orthogonal electric and magnetic fields. There exists a specific energy density within electromagnetic waves and it is the sum of the energy density contained within the mutually perpendicular electric [E_y(x,t) = E₀ cos(kx – ωt)] and magnetic fields [B_x (x, t) = B₀ cos(kx – ωt)] and is given as à 

u(x,t) = 0.5* ε*0 E² + 0.5 * (1/m₀) * B²

Another major categorization of waves derives from the differences in motion of different kinds of waves. Classified into transverse and longitudinal, these two categories are the subject of discussion in this article.

• Transverse motion in waves causes the propagation medium to oscillate or vibrate perpendicular to the direction of the wave motion.

Crests/peaks and troughs/valleys develop in the medium as it experiences periodic displacement at right angles to the direction of the waves; propagation. Ripples on water surface & their deadlier versions, tsunamis, secondary(S) or shearing seismic waves, and electromagnetic waves exhibit transverse wave motion.

• Longitudinal waves cause their propagation medium to vibrate or oscillate along the direction of the wave’s movement.

Given the periodic nature of wave motion, longitudinal waves propagate and transfer energy by compressing and rarefying the propagation medium. Sound and ultrasound waves, vibrations in spring, and primary(P) or compressional seismic waves propagate using longitudinal wave motion.

Now, let’s look at the two wave motion variations.

Transverse Waves

As we mentioned, transverse waves are sets of all those waves that propagate energy by oscillating the medium particles orthogonally to the direction of wave motion. One of the most powerful examples of transverse waves is the secondary (S) or shearing seismic waves produced during earthquakes that cause rock particles to slide up & down and produce immense amounts of shearing stress.

Other prominent examples of transverse wave motion are à 

• Elastic-plastic waves propagating through a string
• Any electromagnetic waves like light, radio waves, X-rays, microwaves, gamma rays, ultraviolet rays (due to orthogonally oscillating electric & magnetic fields)
• Waves are produced by shaking a rope tied at the other end.
• Shearing/secondary (S) seismic waves
• Mexican waves in a stadium

Properties & Behaviour of Transverse Waves

The mutually orthogonal oscillation of medium particles concerning the wave motion produces sinusoidal waveforms within the medium. As we all know, the periodic nature of waves lends certain harmonics properties to the phenomenon that are depicted through certain key parameters, namely, amplitude, frequency, and wavelength.

Frequency and wavelength together determine the characteristics of any kind of wave, irrespective of the motion or wave type, and the same goes for transverse waves.

• The frequency of a transverse wave is the number of oscillations that pass through a particular point in the medium in one second. So, if the frequency of a transverse wave is 50 Hertz, then it means that 50 wave oscillations pass through a medium segment in one second.
• The wavelength of a transverse wave is the length of one complete wave oscillation. As transverse waves propagate by causing medium particles to vibrate perpendicular to the direction of motion, they are best depicted using sinusoidal waveforms. And, in the case of sinusoids, a complete wavelength is the length between a complete peak/crest and a complete valley/trough.
• The product of the frequency and wavelength of the wave gives the velocity of any non-dispersive waveform. So, if the frequency and wavelength of a transverse wave are f and l, respectively, then its velocity is given by v = fl.

For transverse waves, the direction of the wave’s displacement is linearly polarised, while the magnitude of the displacement is sinusoidal.

• The equation of a wave depicts key aspects of the wave, such as amplitude, the nature of the wave function, the phase, the angular velocity, and the motion of the medium as the wave is traversing.

The above is true for any progressive wave, transverse or longitudinal. For a transverse sinusoidal wave, the equation is given as

Y_y(x,t) = a sin (kx -ωt + F)

Where Fdepicts an argument for the sine function that turns the equation into a linear combination of sine and cosine functions, a is the wave amplitude, k is the wave number or the inverse of the wavelength, and ω is the angular or harmonic velocity.

The above wave equation depicts how medium particles displace and oscillate. Differentiating the above equation gives us the PARTICLE VELOCITY of transverse waves.

Both transverse and longitudinal waves are traveling waves, and both can be depicted using sinusoidal functions if the waveform is sinusoidal. The nature of waveforms, however, depends upon the nature of the energy and the medium.

Transverse Wave Propagation

• The amplitude depicts the extent of oscillations of medium particles. As the wave traverses, the displacement of the wave varies between +aand -a.
• The speed of mechanical waves depends on the mechanical properties of the medium they are traversing.

Like any harmonic or oscillating motion, wave motion involves restoring force and stabilising the medium particles once the wave energy has moved on. If we consider transverse waves travelling through a string, then two forces contribute to the wave’s motion through the string, à, the tension of the string T (the restoring force) and the mass density of the string m (the inertial property of the string).

The wave velocity, in terms of these two forces, is then given as

V= Ö(T/ m)

Note how the velocity of a transverse mechanical wave does not depend on the waveform, the frequency, or the amplitude. This is the WAVE VELOCITY of transverse and is independent of time.

However, high-velocity waves do depend substantially on their frequencies.

Now it is time to look at the nature and features of longitudinal waves.

Longitudinal Waves

As mentioned, longitudinal waves traverse by compressing and rarefying the propagation medium. The medium particles oscillate here, too, as expected from periodic motions; only in this case, the oscillations are in the direction of the propagation of the wave. However, as in the case of transverse waves, only the energy propagates from one point to another, not the medium.

Some of the most prominent examples of longitudinal waves in real life are:

• Sound waves
• Ultrasonic waves
• Pressure waves
• Primary (P) or Compression seismic waves
• Waves travelling through a spring

Properties & Behaviour of Longitudinal Waves

• Longitudinal waves have medium displacements in parallel to the wave displacement.
• Waveforms can be sinusoidal, and in such cases, longitudinal waveforms can be depicted using the sinusoidal wave equation, just like the one we used to depict a transverse sinusoidal wave.

Sinusoids are considered the purest of all waveforms, and sinusoidal longitudinal waves can be depicted with the following equation à 

Y_x(x,t) = a sin (kx -ωt + F)

The parameters and variables remain the same. As the medium particles vibrate or oscillate parallelly along the wave, the y(x,t) depicts the horizontal motion of the medium particles.

Differentiating the above equation gives us the particle velocity of longitudinal waves à 

V_x (x, t) = ∂y(x,t)/∂t=−Aωcos(kx−ωt+φ).

Longitudinal Wave Propagation

• Like transverse waves, the wave velocity of longitudinal waves does not depend on time. The speed depends on the elastic and inertial properties of the medium.

If we consider the speed of sound through a solid medium, the compressions and rarefactions depend on the mass and elastic properties. The compressional stress and strain are determined by the Bulk modulus of the medium and are given as à 

B = -[ΔP/(ΔV/V)]

ΔP is the change in pressure that creates all the compressive stress, while ΔV/V is the volumetric strain generated due to the stress.

If we consider compressive stress to be the vital force, then the mass density of the medium is the other key factor for longitudinal mechanical waves. Bulk modulus has the same dimensions as restorative stress, and thus, the longitudinal wave velocity can be calculated as à 

V= Ö(B/ρ)

Where ρ is the mass density of the medium and provides the countering inertial property.

Now that we have mulled over the key features, properties, and behaviour of both transverse and longitudinal waves, let’s pit one against the other.

Comparing & Contrasting Between The Two Motions

Apart from the direction and nature of displacement of medium particles, there are not many differences between transverse & longitudinal waves. If the underlying waveform is sinusoidal, the wave equations are the same in both cases, with only the nature of displacements being different.  

Here are the biggest differences between transverse and longitudinal waves.

Wave motion direction

Transverse Waves à The direction of wave motion is perpendicular or orthogonal to the direction of particle oscillations.

Longitudinal Wavs à  The direction of wave propagation is parallel or in the same direction as that of medium particle oscillations.

Particle motion direction

Transverse Waves à Particle displacements are perpendicular to the direction of wave motion.

Longitudinal Waves à Particles oscillate in the direction of wave propagation through compressions and rarefactions by changing the mass density of the medium particle.

Waveform and Oscillation

Both transverse and longitudinal waveforms can be sinusoidal in nature, as we saw in the sections above. The exact size and shape of any waveform depend on the source of the energy or disturbance and the nature of the waveform.

Oscillations of medium particles depend on the type of wave motion. Transverse wave motions make medium particles oscillate orthogonally to the direction of wave propagation. On the other hand, longitudinal waves cause medium particles to vibrate or displace in the direction of the wave.

Energy Transfer

The energy and power transferred by a wave do NOT depend upon the nature of the wave motion. Amplitude and frequency are the factors that determine the energy contained in a wave. Wave equations contain both these parameters and depict discrete packets of energy.

Waves with high amplitudes cause the medium particles to vibrate at higher magnitudes. If the wave frequency is high, the medium undergoes massive vibrations more frequently. The amount of energy transferred through the medium is thus higher if amplitudes and frequencies are higher.

The total mechanical energy of any wave is the total of its kinetic and potential energies.

• If we consider a string vibrator (a device that vibrates a string), then the kinetic energy of each minute mass element of a string as it undergoes oscillations is à

ΔK=0.5* (μΔx) v_y²

where μ  is the mass density, Δx is the mass element, ΔK is the kinetic energy of that element, and v_y  is the mass velocity. The wave traversing through the string is undergoing transverse motion.

For longitudinal wave motion, only the mass or particle velocity direction changes.

• The velocity of the mass element is the particle velocity and is obtained by differentiating the wave equation. Thus, we have à

dK = (0.5) * (μdx)* A²ω²cos²(kx)

If we integrate across the entire length of the string, the total kinetic energy due to the wave comes to à 

K_l = (1/4) * A^2*ω²* μ*l

Where lis the length of the entire string.

• The potential energy of the oscillating string is also given by à.

U_l = (1/4) * A^2*ω²* μ*l

(Can you find out how and why the kinetic and potential energy of the string have the same expression?)

• Hence the total mechanical energy of the vibrating string is à

E_l-= (0.5) * A^2*ω²* μ*l

The time-average power of a sinusoidal transverse mechanical wave can be calculated by taking a specific reference point and then dividing the equation above by the time taken by the wave to cross that specific point.

In Conclusion

Let’s wrap up things up with a quick reiteration of everything discussed.

• Longitudinal waves have medium particles displacing in the direction of the wave’s propagation. They oscillate & transfer energy through compressions and rarefactions.

The oscillations can be determined using a specific wave equation. If the waveform of those oscillations is sinusoidal, then the waveform equation is given as

Y_x(x,t) = a sin (kx -ωt + F)

Energy propagates through differential compressive pressure across the body of the medium. The amplitude of the above equation depicts the distance between the medium particles during compression. The closer they are, the higher the amplitude of the waveform.

• Transverse waves cause medium particles to oscillate orthogonal to the wave motion. Particles do not transfer energy through compressions and rarefactions but by oscillating up & down with a certain amplitude.

Transverse waves have crests & troughs that are the counterparts of compressions and rarefactions in longitudinal waves.

• The study of wave motions and particle & wave velocities are fundamental in electronics communication, telecommunications, telemetry, material science, seismography, civil engineering, material engineering, structural, automobile & aeronautical engineering, cosmology, optics, and numerous other science & engineering domains.

If studying waves and motion seems like a struggle, you need to put in more effort. Solid advanced mathematical concepts and a clear idea of mechanics are essential to understand and master every aspect of waves and wave motion.

Put in the effort, and go through definitions & examples from different sources. solve as many sums as possible. If things still seem problematic, connect with the subject matter experts of MyAssignmenthelp.com.